Learning Objectives
Introduction
Tutorial
If n is even:
If n is even and a is negative, then the root is not a real number.
If n is odd:
When there is no index number, n, it is understood to be a 2 or square root. For example:
= principal square root of x.
Note that NOT EVERY RADICAL is a square root.
If there
is an index number n other than the
number
2, then you have a root other than a square root.
Example 1: Evaluate or indicate that the root is not a real number.
What do you think it is?
Let’s find out if you are right:
Since 10 squared is 100, 10 is the square root of 100.
Note that we are only interested in the principal root and since 100 is positive and there is not a sign in front of the radical, our answer is positive 10. If there had been a negative in front of the radical our answer would have been -10.
Example 2: Evaluate or indicate that the root is not a real number.
What do you think it is?
Let’s find out if you are right:
Since 2 raised to the fourth power is 16 and we are negating that, our answer is going to be -2.
Note that the negative was on the outside of our even radical. If the negative had been on the inside of an even radical, then the answer would be no real number.
Example 3: Evaluate or indicate that the root is not a real number.
What do you think it is?
Let’s find out if you are right:
Since there is no such real number that when we square it we get -100, the answer is not a real number.
If n is an even positive
integer, then
If n is an odd positive integer, then
Example 4: Simplify .
Since the root number and the exponent inside are equal and are the even number 2, we need to put an absolute value around y for our answer.
The reason for the absolute value is that we do not know if y is positive or negative. So if we put y as our answer and it was negative, it would not be a true statement.
For example if y was -5, then -5 squared would be 25 and the square root of 25 is 5, which is not the same as -5. The only time that you do not need the absolute value on a problem like this is if it stated that the variable is positive.
Example 5: Simplify .
This time our root number and exponent were both the odd number 3. When an odd numbered root and exponent match then the answer is the base whether it is negative or positive.
We can use the product rule of radicals (found below) in reverse to help us simplify the nth root of a number that we cannot take the nth root of as is, but has a factor that we can take the nth root of. If there is such a factor, we write the radicand as the product of that factor times the appropriate number and proceed.
We can also use the quotient rule of radicals (found below) to simplify a fraction that we have under the radical.
Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth. I will be using that phrase in some of the following examples.
A Product of Two RadicalsNote that if you have different index numbers, you CANNOT multiply them together.
Also, note that you can use this rule in either direction depending on what your problem is asking you to do.
Example 6: Use the product rule to simplify .
Since we cannot take the cube root of 15 and 15 does not have any factors we can take the cube root of, this is as simplified as it gets.
Example 7: Use the product rule to simplify .
Since we cannot take the fourth root of what is inside the radical sign and 24 does not have any factors we can take the fourth root of, this is as simplified as it gets.
Example 8: Use the product rule to simplify .
Check it out:
*Use the prod.
rule of radicals to rewrite
*The square root of 25 is 5
Example 9: Use the product rule to simplify .
Check it out:
*Use the prod.
rule of radicals to rewrite
*The cube root of is
If n is even, x and y represent any nonnegative real number and y does not equal 0.
If n is odd, x and y represent
any real number and y does not equal 0.
This rule can also work in either direction.
Example 10: Use the quotient rule to simplify .
*The cube root of -1 is -1 and
the cube root
of 27 is 3
Example 11: Use the quotient rule to simplify .
*Use the quotient rule of radicals to rewrite
*Simplify the fraction
*Use the prod. rule of radicals to rewrite
*The square root of 4 x squared
is 2|x|
The following are two examples of two different pairs of like radicals:
You add or subtract them in the same fashion that you do like terms. Combine the numbers that are in front of the like radicals and write that number in front of the like radical part.
Example 12: Add .
Example 13: Subtract .
Can you think of what that factor is?
Let’s see what we get when we simplify the second radical:
*Square root of 25 is 5
Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth.
Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.
Example 14: Rationalize the denominator .
Square roots are nice to work with in this type of problem because if the radicand is not a perfect square to begin with, we just have to multiply it by itself and then we have a perfect square.
So in this case we can accomplish this by multiplying top and bottom by the square root of 5:
*Den. now has a perfect square
under sq. root
AND
Example 15: Rationalize the denominator .
So in this case, we can accomplish this by multiplying top and bottom by the cube root of :
*Mult. num. and den. by cube
root of
*Den. now has a perfect cube under cube root
AND
*Cube root of 8 a cube is 2a
Also, we cannot take the cube root of anything under the radical. So, the answer we have is as simplified as we can get it.
Because we now have two terms, we are going to have to approach it differently than when we had one term, but the goal is still the same.
a + b and a - b are conjugates of each other.
When you multiply conjugates together you get:
Example 16: Rationalize the denominator .
So what would the conjugate of our denominator be?
It looks like the conjugate is .
*Use distributive prop. to
multiply the numerators
*In general, product of conjugates
is
AND
*Square root of 3 squared is 3
Example 17: Rationalize the denominator .
So what would the conjugate of our denominator be?
It looks like the conjugate is .
*Use distributive prop. to
multiply the numerators
*In general, product of conjugates
is
AND
*Divide BOTH terms of num. by -2
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1d: Evaluate or indicate that the root is not a real number.
Practice Problems 2a - 2b: Use the product rule to simplify the expression.
Practice Problems 3a - 3b: Use the quotient rule to simplify the expression
Practice Problems 4a - 4b: Add or subtract.
Practice Problems 5a - 5b: Rationalize the denominator.
Need Extra Help on these Topics?
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut39_simrad.htm
This webpage covers how to multiply, divide, and simplify radical
expressions.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut40_addrad.htm
This webpage helps you with adding and subtracting like radicals.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut41_rationalize.htm
This webpage explains how to rationalize denominators.
http://www.purplemath.com/modules/radicals.htm
This website goes over radicals and rationalizing denominators.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 6, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.